A secondary motivation comes from the recent study
(see [1]} and [2]}) of the so-called Poisson
cylinder model.
This is a model where the percolation objects are infinitely long,
and while it was shown in [1]} that the vacant set undergoes
a phase transition, it was later shown in [2]} that the occupied
component does not. (We point out that, since the cylinders are unbounded,
the phase transition in question is not whether there exists unbounded
connected components, but rather whether all cylinders in the
model belong to the same connected set.)
It is natural to think of the Poisson cylinder model as a limit of the
Poisson stick model as the length \(L\)  of the sticks diverges.
From this viewpoint it then becomes natural to ask
how the percolation threshold behaves in this limit.
